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[2] See L. S. Schulman, Techniques and Applications of Path Integrals (Wiley, New York, 1981), Chaps. 22, 23, and 31, for a discussion of some of the complexities. See also Part III of Assa Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, New York, 1994).

[3] L. D. Faddeev, in Methods in Field Theory, Les Houches 1975, edited by R. Balian and J. Zinn-Justin (NorthHolland, Amsterdam, 1976).

[4] J. R. Klauder, Phys. Rev. D 19, 2349 (1979), and in Path Integrals, and Their Applications in Quantum, Statistical, and Solid State Physics, edited by G. J. Papadopoulos and J. T. Devreese (Plenum, New York, 1978).

[5] It should be stressed that our remarks apply only to tunneling, i.e., to infinite time propagators. Paths with nonclassical endpoints are known to be important for finite time problems, e.g., Larmor precession in the semiclassical approximation. In these cases, however, these are the only paths that contribute. There is no problem of which we are aware in which both types of paths, with and without jumps, need to be included.

[6] H. J. Lipkin, N. Meshov, and A. J. Glick, Nucl. Phys. 62, 188 (1965), R. Shankar, Phys. Rev. Lett. 45, 1088 (1981), W. G. Harter and C. W. Patterson, J. Chem. Phys. 80, 4241 (1984), I. Hamamoto and B. Mottelson, Phys. Lett. B 333, 294 (1994); Phys. Scr. T56, 27 (1995).

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[8] M. Stone, K.-S. Park, and A. Garg, J. Math. Phys. 41, 8025 (2000).

[9] In this paper we only consider external fields along ˆz. Also, note that one can add a term proportional to J · J to H, since that commutes with H. This can change the 2 superficial appearance of H and the coefficients of Jx , etc. The physics, of course, is unaffected.

[10] A. L. Barra, P. Debrunner, D. Gatteschi, C. E. Schulz, R. Sessoli, Europhys. Lett. 35, 133 (1996);